Definition of Lebesgue measure on the Circle $X=\mathbb{R}/\mathbb{Z}$.

Question

I was given the following definition:

I feel this is not adequate and would only define Lebesgue on $[0,1]$? (Wikipedia uses a pushforard of the complex exponential) Also later I am given a proof that rotation by $\alpha$, $T:X→ X, T(x) = x+\alpha \mod 1$ is $\mu$-invariant(Lebesgue on circle), but is it enough?

This seems to treat the circle as $[0,1]$ at times which is confusingly inconsistent.

Answer 1

I think that the definition is fine. Given an interval $[a, b]$ we have two cases. Either $b-a >=1$ and $\mu([a,b]/\sim) = \mu(X) = 1$, or $0< b-a < 1$ and $\mu([a, b]) = b-a$.

Now, you might wish that the definition treat the case of intervals of the form $[3/4, 1/4]$. To that I would pedantically say something like "do you mean an image of $[3/4, 5/4]$ or of $(-\infty, 1/4]\cup [3/4, \infty)$ or what?"

Then we get to the proof of rotational invariance. Again, I think its fine, so long as we realise that $(a, b)$ is perfectly good description of a subset of $X$ regardless of where $a$ and $b$ are compared to $0$, $1$ or any other integer.